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2. WARM-UP 1.2 Evaluate each expression. • 32 – 2(4 – 2) • [(8 + 3) · 3 – 12] ÷ 7 Evaluate each expression if a = 0.5, b = 6, and c = - 3 • ac – bc + a • a(b + c) + ab2 • Find the interest rate I if I = prt and p = 2500, r = 8%, and t = 3.

3. WARM-UP 1.2 Evaluate each expression. • 32 – 2(4 – 2) 5 • [(8 + 3) · 3 – 12] ÷ 7 3 Evaluate each expression if a = 0.5, b = 6, and c = - 3 • ac – bc + a 17 • a(b + c) + ab2 19.5 • Find the interest rate I if I = prt and p = 2500, r = 8%, and t = 3. \$600

4. 1.2 Properties of Real Numbers Algebra 2 Mrs. Spitz Fall 2006

5. 27 13 97 - 7 3/5 88 45 - 8 71/8 12 17/3 - 65/18 206 73/4 Answers to problems 1.1

6. 1.2 Objectives: • Determine sets of numbers to which a number belongs. • Use the properties of real numbers to simplify expressions.

7. Introduction • All the numbers we use in everyday life are real numbers. Each real number corresponds to exactly one point on the number line, and every point on the number line represents exactly one real number.

8. m n Introduction • Every real number can either be classified as either rational or irrational. A rational number can be expressed as a ratio , where m and n are integers and n is not zero. The decimal form of a rational number is either a terminating or repeating decimal. Some examples of rational numbers are ⅔, 1.23, 5.8, -7, and 0.

9. Introduction • Any real number that is not rational is irrational. √2, , and √7 are irrational numbers. • The sets of natural numbers {1, 2, 3, 4, 5, . . . } • Whole numbers {0, 1, 2, 3, 4, . . . } • Integers {. . . -2, -1, 0, 1, 2, . . . } are subsets of the rational numbers.

10. The Venn Diagram to the right shows the relationship between all of these sets of numbers. COPY THIS!!! Venn Diagram

11. √6 3/8 -9 Example 1: Name the sets of numbers to which each number belongs. • Irrationals, reals • Rationals and reals • Integers, rationals, reals

12. √9 6 ÷ 10 Example 2: Evaluate each expression. Then name the sets of numbers to which each value belongs. • √9 = 3; so natural numbers, whole numbers, integers, rationals, reals • 6 ÷ 10 = .6 or 3/5 so rational numbers and real numbers

13. Operations with real numbers have several important properties. One of the basic properties of addition and multiplication is commutativity. The order in which two real numbers are added or multiplied does not change their sum or their product. 5 + 9 = 9 +5 14 = 14 Or 12 · 4 = 4 · 12 48 = 48 Notes: Commutative Property of Addition/Multiplication

14. Another basic property of addition and multiplication is associativity. The way three or more real numbers are grouped, or associated, does not change their sum or product. (7 + 6) +9 = 7 + (6 + 9) 13 + 9 = 7 + 15 22 = 22 Or (10 · 5) · 2 = 10 · (5 · 2) 50 · 2 = 10 · 10 100 = 100 Notes: Associative Property of Addition/Multiplication

15. The sum of any real number and 0 is the original number. So, for real numbers, the additive identity is 0. 6.7 + 0 = 6.7 Or 0 + √3 = √3 Notes: Additive Identity

16. Each real number has a unique additive inverse or opposite. The sum of a number and its opposite is ALWAYS 0. 8 + (-8) = 0 Or -⅓ + ⅓ = 0 Notes: Additive inverse or opposite

17. The Multiplicative Identity for real numbers is ALWAYS 1 since the product of any real number and 1 is the original number 5/8 · 1 = 5/8 Or (1)(3.9) = 3.9 Notes: Multiplicative identity

18. Each real number, except 0, has a unique multiplicative inverse or reciprocal. The product of a real number and its reciprocal is ALWAYS 1. ⅛(8) = 1 (-0.2)(-5) = 1 Reminder: A number and its reciprocal have the same sign. Notes: Multiplicative inverse or reciprocal

19. Summary Note: -a is read “the opposite of a.”

20. Ex. 3 • A fast-food restaurant offered a special on their Biggie Burger to entice customers at lunch time. The 99¢ special ran Monday through Friday. The number of Biggie Burgers sold each day are recorded in the table. What was the average amount received from the sale of Biggie Burgers during each day?

21. Ex. 3 • An average is calculated by dividing the total amount by the number of items. In this case, it would be the total dollar amount divided by the number of days, 5. There are two ways to find the total dollar amount.

22. Ex. 3 • T = 0.99(246 + 303 + 182 + 341 + 378) T = 243.54 + 299.97 + 180.18 + 337.59 + 374.22 T = 1435.50 (2) T = 0.99(246 + 303 + 182 + 341 + 378) T = 0.99(1450) T = 1435.50 Now find the average by dividing the total by 5. 1435.50 ÷ 5 = 287.10 The average daily revenue received from Biggie Burgers was \$287.10

23. Distributive Property • This is an application of a property that involves both addition and multiplication. It is called the distributive property. • For all real numbers a, b, and c, a(b + c) = ab + ac and (b + c)a = bc + ba

24. 5(3m – 7n) + 3(4m + n) = 5(3m) -5(7n) +3(4m) + 3(n) =15m – 35n + 12m + 3n =(15m + 12m) + (-35n + 3n) = 27m – 32n Write the expression. Distributive Property Multiply Collect like terms Simplify Ex. 4:Simplify 5(3m – 7n) + 3(4m + n)

25. Assignment • pp. 16-17 #6-48 all • SHOW ALL WORK OR ELSE IT DOESN’T COUNT. Copying somebody else’s work won’t work on the ACT. Colleges already don’t trust the high school grading system, and they make their own tests. You must test after this year to get into the next higher math class. • Please see me if you are having the slightest problem doing the homework. It may be you need to transfer to CORD, so you can get the math credit or start tutoring so we can get you the extra help you need.